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1.
5
7
= y13
y( y )( y )
2.
(2d )(5d )
3.
(2a b ) (3a b)
3
7 4 3
4
= -10d7
6
2
= – 72a33b14
4.)
(40a b )
6 2
5a b
3 7 0
1
6 2
5a b
5.)
6
2x 4
( 5 )
y
24
16 x
20
y
6.)
4 0
a ba
1
2
a
2
Algebra 1 ~ Chapter 8.4
Polynomials
Remember: A monomial is a number, a
variable, or a product of numbers and
variables with whole-number exponents.
“Mono” – single term
The degree of a monomial is the sum of
the exponents of the variables. A
constant has degree 0.
Ex. 1 - Find the degree of each
monomial.
A. 4p4q3
The degree is 7.
B. 7ed
The degree is 2.
Add the exponents of the variables:
4 + 3 = 7.
A variable written without an
exponent has an exponent of 1.
1+ 1 = 2.
C. 3
The degree is 0.
There is no variable, but you can
write 3 as 3x0.
* A polynomial is the sum or difference of
monomials. The degree of a polynomial is the
degree of the term with the greatest degree.
“poly” – many
An example of a polynomial is 3a + 4b – 8c
That expression consists of three monomials
“combined” with addition or subtraction.
Some polynomials have special names based
on the number of terms they have.
Ex. 2 – Find the degree of each polynomials.
Then name the polynomials based on #
of terms.
A.) 5m4 + 3m
This polynomial
The greatest degree is 4, so the
degree of the polynomial is 4.
B.) -4x3y2 + 3x2 + 5
The degree of the
polynomial is 5.
C.) 3a + 7ab – 2a2b
The degree of the
polynomial is 3.
has 2 terms, so
it is a binomial.
This polynomial
has 3 terms, so
it is a trinomial.
This polynomial
has 3 terms, so
it is a trinomial.
Writing Polynomials in Order

The terms of a polynomial are usually
arranged so that the powers of one variable
are in ascending (increasing) order or
descending (decreasing) order.
Ex. 3 – Arrange the terms of the polynomial so
that the powers of x are in descending order.
6x – 7x5 + 4x2 + 9
Find the degree of each term. Then arrange them in
decreasing order:
6x – 7x5 + 4x2 + 9
Degree
1
5
2
–7x5 + 4x2 + 6x + 9
0
5
2
1
0
The polynomial written in descending
order is
-7x5 + 4x2 + 6x + 9.
Ex. 4 - Write the terms of the polynomial so that the
powers of x are in descending order.
y2 + y6 − 3y
Find the degree of each term. Then arrange them in
decreasing order:
y2 + y6 – 3y
Degree
2
6
1
y6 + y2 – 3y
6
2
1
The polynomial written in descending order is
y6 + y2 – 3y.
6-2 Adding and Subtracting Polynomials
Algebra 1 ~ Chapter 8.5
“Adding and Subtracting Polynomials”
6-2 Adding and Subtracting Polynomials
Warm Up - Simplify each
expression by combining like terms.
1. 4x + 2x
6x
2. 3y + 7y
10y
3. 8p – 5p
3p
4. 5n + 6n2
Not like terms
9x2
5. 3x2 + 6x2
6. 12xy – 4xy
8xy
o Just as you can perform operations on
numbers, you can perform operations on
polynomials.
o To add or subtract polynomials,
combine like terms.
Example 1: Adding and Subtracting
Monomials
A. 12p3 + 11p2 + 8p3
12p3
+
8p3
+
11p2
20p3 + 11p2
B. 5x2 – 6 – 3x + 8
5x2 – 3x + 8 – 6
5x2 – 3x + 2
Arrange the terms so the
“like” terms are next to each
other and then simplify.
Polynomials can be added in either
vertical or horizontal form.
Simplify (5x2 + 4x + 1) + (2x2 + 5x + 2)
In vertical form, align the like terms and add:
2
5x 2
2x
+ 4x + 1
+
+ 5x + 2
2
7x + 9x + 3
In horizontal form, regroup and combine like terms.
(5x2 + 4x + 1) + (2x2 + 5x + 2)
= (5x2 + 2x2) + (4x + 5x) + (1 + 2)
= 7x2 + 9x + 3
Example 2: Adding Polynomials
A. (4m2 + 5m + 1) + (m2 + 3m + 6)
(4m2 + 5m + 1) + (m2 + 3m + 6)
(4m2 + m2) + (5m + 3m) + (1 + 6)
5m2 + 8m + 7
B. (10xy + x) + (–3xy + y)
(10xy + x) + (–3xy + y)
(10xy – 3xy) + x + y
7xy + x + y
Example 2: Adding Polynomials
C.
Subtracting Polynomials
Simplify (4x + 5) – ( 2x + 1)
Option #1:
Option #2: Recall that you
(4x – 2x) + (5 – 1 )
can subtract a number by
adding its opposite.
2x + 4
(4x + 5) + (-2x – 1)
(4x + -2x) + (5 + -1)
2x + 4
Example 3: Subtracting Polynomials
A. (4m2 + 5m + 1) − (m2 + 3m + 6)
(4m2 + 5m + 1) − (m2 + 3m + 6)
(4m2 − m2) + (5m − 3m) + (1 − 6)
3m2 + 2m – 5
B. (10x3 + 5x + 6) − (–3x3 + 4)
(10x3 - - 3x3) + (5x – 0x) + (6 – 4)
13x3 + 5x + 2
Example 3C: Subtracting Polynomials
(7m4 – 2m2) – (5m4 – 5m2 + 8)
(7m4 – 5m4) + (−2m2 – −5m2) + (0 – 8)
(7m4 – 5m4) + (–2m2 + 5m2) – 8
2m4 + 3m2 – 8
Example 3D: Subtracting Polynomials
(–10x2 – 3x + 7) – (x2 – 9)
(–10x2 – x2) + (−3x – 0x) + (7 – -9)
–11x2 – 3x + 16
Lesson Wrap Up
Simplify each expression.
1. 7m2 + 3m + 4m2
11m2 + 3m
2. (r2 + s2) – (5r2 + 4s2)
–4r2 – 3s2
3. (10pq + 3p) + (2pq – 5p + 6pq)
4. (14d2 – 8) – (6d2 – 2d + 1)
18pq – 2p
8d2 +2d – 9
5. (2.5ab + 14b) – (–1.5ab + 4b)
4ab + 10b
Assignment
Study Guide 8-4 (In-Class)
 Study Guide 8-5 (In-Class)
 Skills Practice 8-4 (Homework)
 Skills Practice 8-5 (Homework)
